The sensitivity index or discriminability index or detectability index is a dimensionless statistic used in signal detection theory. A higher index indicates that the signal can be more readily detected. Definition The discriminability index is the separation between the means of two distributions (typically the signal and the noise distributions), in units of the standard deviation. Equal variances/covariances For two univariate distributions and with the same standard deviation, it is denoted by ('dee-prime'): . In higher dimensions, i.e. with two multivariate distributions with the same variance-covariance matrix , (whose symmetric square-root, the standard deviation matrix, is ), this generalizes to the Mahalanobis distance between the two distributions: , where is the 1d slice of the sd along the unit vector through the means, i.e. the equals the along the 1d slice through the means. This is also estimated as . Unequal variances/covariances When the two distributions have different standard deviations (or in general dimensions, different covariance matrices), there exist several contending indices, all of which reduce to for equal variance/covariance. Bayes discriminability index This is the maximum (Bayes-optimal) discriminability index for two distributions, based on the amount of their overlap, i.e. the optimal (Bayes) error of classification by an ideal observer, or its complement, the optimal accuracy : , where is the inverse cumulative distribution function of the standard normal. The Bayes discriminability between univariate or multivariate normal distributions can be numerically computed (Matlab code), and may also be used as an approximation when the distributions are close to normal. is a positive-definite statistical distance measure that is free of assumptions about the distributions, like the Kullback-Leibler divergence . is asymmetric, whereas is symmetric for the two distributions. However, does not satisfy the triangle inequality, so it is not a full metric. In particular, for a yes/no task between two univariate normal distributions with means and variances , the Bayes-optimal classification accuracies are: , where denotes the non-central chi-squared distribution, , and . The Bayes discriminability can also be computed from the ROC curve of a yes/no task between two univariate normal distributions with a single shifting criterion. It can also be computed from the ROC curve of any two distributions (in any number of variables) with a shifting likelihood-ratio, by locating the point on the ROC curve that is farthest from the diagonal. For a two-interval task between these distributions, the optimal accuracy is ( denotes the generalized chi-squared